Divide and express the result in standard form. 8i/(4 - 3i)

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Identify the problem: You need to divide the complex number \$8i$ by the complex number $(4 - 3i)$ and express the result in standard form, which is $a + bi$ where $a$ and $b$ are real numbers.

To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator. The conjugate of $(4 - 3i)$ is $(4 + 3i)$. So multiply both numerator and denominator by $(4 + 3i)$:

\[\frac{8i}{4 - 3i} \times \frac{4 + 3i}{4 + 3i}\]

Use the distributive property (FOIL) to expand both the numerator and the denominator:

- Numerator: $8i \times (4 + 3i)$

- Denominator: $(4 - 3i)(4 + 3i)$

Simplify the denominator using the difference of squares formula: $(a - bi)(a + bi) = a^2 + b^2$. Here, $a=4$ and $b=3$, so the denominator becomes $4^2 + 3^2$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Complex Number Standard Form

The standard form of a complex number is expressed as a + bi, where a and b are real numbers, and i is the imaginary unit with i² = -1. Writing complex numbers in this form separates the real and imaginary parts clearly.

Division of Complex Numbers

Dividing complex numbers involves multiplying the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part in the denominator. This process simplifies the expression into a standard form.

Complex Conjugate

The complex conjugate of a number a + bi is a - bi. Multiplying a complex number by its conjugate results in a real number, which helps in rationalizing denominators when dividing complex numbers.

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